Game luck will affect your chances of

• making a monster peaceful by untrapping it
• making a (too heavy to lift) monster peaceful by trying to untrap it ("<monster> thinks it was nice of you to try.")
• getting an alignment bonus for untrapping a monster
• accidentally setting off a landmine or beartrap that you are trying to set
• having monsters get mad when they are trapped by one of your traps
• setting off a trap you are trying to disarm
• disarming a shooting trap
• having chest traps fail to work (flame fizzles out, etc)
• having chest traps that do work limited to relatively minor effects (especially if you have a unicorn horn)
  
  
• getting any result from prayer (except upset gods---Luck < 0 causes this)
• getting help from your god when poorly aligned
• getting a good result from prayer
• getting the full range of bonus results from prayer
• being confronted by a minion of the god of the altar you converted (or tried to)
• causing the gods to be angry when praying in hell
• having angry gods being less fire and brimstone in their approach
  
  
• hitting a monster
• hitting a monster with a thrown object
• hitting a monster with a iron ball when it jerks you back (?)
• hitting something with a boomerang without it splintering
  
  
• getting good things from thrones (Luck < 0 is sometimes bad, Luck > 0 is sometimes good)
• destroying/getting gold and gems from throne (Luck >= 5 maximises number of gems)
• getting good things from a magic fountain (natural luck >= 0 required; natural luck >= 4 desirable)
• finding secret doors, corridors and traps.
• breaking open a door
• opening a door
  
  
• having your objects resist water damage
• blessed metal items resisting rust
  
  
• having a wand of wishing succeed (Luck >= 0 guarantees this)
• getting what you wish for (Luck >= 0 is pretty much essential)
  
  
• writing a scroll you don't know
• picking something off the floor by applying your whip
• attempting a teleport spell in need (guaranteed if Luck >= 1)
• getting at least 2 identifies from a blessed scroll of identify (Luck > 0)
• having polymorphed objects destroyed
• having a split off long worm remain tame/peaceful

So, yes, it affects your fighting ability, but it does not affect your dodging ability nor your resistance to magic.

Is there a cap on how high luck can go?

Yes. Natural luck ranges from -10 to 10. A luckstone will give you either extra luck (an additional +3) or reduced luck (-3). A comment claims that on moonlit night, the maximum for natural luck is 11, but I don't believe it---all that happens is a routine is called to add one to your luck but which will not exceed 10 afterwards.

How much luck is worthwhile?

The higher the better for some things (like hitting a monster). But you definitely want it to be positive to avoid nasty things like having your wand of wishing tell you "Unfortunately, nothing happens."